Can You Calculate the Stopping Distance at a Traffic Light?

[ice109]

Homework Statement

(3) A motorist is approaching a green traffic light with speed $v_0$ when the light turns
to amber.
(a) If his reaction time is $\tau$, during which he makes his decision to stop and applies his foot to the
brake, and if the maximum braking deceleration is a, what is the minimum distance $s_{min}$ from the
intersection at the moment the light turns to amber in which he can bring his car to a stop?
(b) If the amber light remains on for a time t before turning red, what is the maximum distance
$s_{max}$ from the intersection at the moment the light turns to amber such that he can continue into the
intersection at speed $v_0$ without running the red light?
(c) Show that if his initial speed $v_0$ is greater than

$v_{0_{max}} = 2a(t- \tau )$

there will be a range of distance from the intersection such that he can neither stop in time nor
continue through the intersection without running the red light.

Homework Equations

kinematics equations

The Attempt at a Solution

A using $vf^2=v_0^2 +2(-a)d$ is $s_{min}=\frac{v_0^2}{2a} +v_0\tau$

B is simply $v_0 t$

C I'm almost clueless. i can sub the given expression into both of my derived equations for $s_{min}$ and $s_{max}$ and all i get is that they both equal $2at(t-\tau)$ which i can't see how to use to prove that $s_{max}<s<s_{min}$.

This is actually an intermediate mechanics question but seems simple enough.

 

[mda]

Substitute your values into the inequality for smax & smin.
Solve for v.

 

[ice109]

the implication needs to go the other way, if p is the statement about velocity and q is the statement about distance i need to prove p -> q, that would prove q->p

even if i did do as you said, how would i "solve" for v?

 

[mike115]

(c) Show that if his initial speed $v_0$ is greater than

$v_{0_{max}} = 2a(t- \tau )$

there will be a range of distance from the intersection such that he can neither stop in time nor
continue through the intersection without running the red light.

s > s_max or s < s_min
v*t > s_max or v*t < s_min

Since we are looking for the maximum initial speed, then
vo_max*t = vo_max^2/(2a) + vo_max*r
t = vo_max/(2a) + r
t - r = vo_max/(2a)
vo_max = 2a(t-r)

 

[ice109]

i don't understand what you've done

why did you equate the two?